Granger-causality tests

Because sometimes it’s more than correlation;
It’s causation
Raquel Acosta RA6977430
Lisa Kustina RA6977197

Used to determine if one time series is
useful in forecasting another
Claims to move beyond correlation to
test for causation
Can only be applied to pairs of variables
May be misleading if true relationship
involves three or more variables

What is causality?
What do you think?

Sir Clive William John Granger
•Born 1943
•Studied applied mathematics and
statistics
•Won Nobel Prize in Economics 2003
•Published a series of articles and
two books analyzing the relationship
between time series
The topic of how to define causality has kept philosophers busy for over two thousand years and has yet to be
resolved. It is a deep convoluted question with many possible answers which do not satisfy everyone, and yet it
remains of some importance. Investigators would like to think that they have found a “cause”, which is a deep
fundamental relationship and possibly potentially useful…

Sir Clive William John Granger
•Born in 1943
•Studied applied mathematics and
statistics
•Won Nobel Prize in Economics 2003
•Published a series of articles and
two books analyzing the relationship
between time series

What is causality?
is all the information in the universe
at time T
F(A|B) is the conditional distribution of A
given B
and are two time series
If this condition holds, then does not cause .
If this condition does not hold, then can be
said to cause because it contains special
information that is not available elsewhere.

1. Run a regression of X
2. Determine a statistically significant time lag
interval
3. Run multiple regressions
1. Make sure additional regressions are also significant
2. Include regressions that add explanatory power
4. Can be repeated for multiple Ys
5. Hopefully, find a clear relationship, such as Y
granger-causes X

Do a series of F-tests on lagged values
of Y to see if those values provide
statistically significant information about
future values of X
Can involve more variables if vector
autoregression is applied to capture
interdependencies

Linear Regression models
Used to describe the dependence of the mean value of one variable
on one or more other variables.
E(Y | x) – the regression of Y on x.
• x is the explanatory variable, or regressor. Y – the mean of variable y.
○
i.e. (y is the occurrences of embezzlement and x is the individuals yearly income)
• Since it’s linear E(Y|x) – β 0 +β 1 x
○
β are our regression coefficents.
○ Y is of unknown variance σ 2
○ Given a sample of n independent pairs of observations (x i
, y i
) the best estimators of β 0
+β 1
is given by the method of Least Squares.
• i.e. minimize Σ(y i - β 0 - β 1 x 1 )
• Gauss-Markov theorem states that the least squares estimator gives the unbiased* estimator of a
parameter having minimum variance.
• It’s good to note the least-squares regression of x on y is not in general the same line as the leastsquares
regression of y on x. – i.e. what is meant by the line of best fit to a data set depends on what
assumptions are made about the nature of any deviations from a fitted line
Goal – select parameters so as to minimize the sum of the squared residuals
• OLS – Ordinary least squares estimation
○ Best linear unbiased estimates if and only if the Gauss-Markov assumptions are satisfied.

Time series models
Standard regression can’t be applied
Common forms
• Autoregressive models
• Moving average models
• Box-Jenkins (Combines these 2)
• ARCH - Autoregressive conditional heteroskedasticity
• GARCH – Generalized ARCH
○ Frequently used for financial time series
• VAR – Vector autoregression
○ Used to understand inter-relationships of variables represented by
systems of equations
Estimates difference equations containing
stochastic components

Autoregressive model
X t = c + φX t-1 + ε t where ε t is white noise with zero
mean and variance σ 2
○ If φ = 1 then X t exhibits a unit root and can be
considered a random walk.
○ For AR of order 1, processes where its parameters φ,
are stationary if φ < 1.
• Let μ be the mean. Then E(X t ) = E(c) + φ E(φX t-1 )
+ E(ε t ) ; so μ = c + φ μ+0
• So var(X t ) = E(X t 2 ) – μ 2 = σ 2 /(1- φ 2 )
• Autocovariance is given by E(X t+n | X t ) - μ 2 =(σ 2 /(1-
φ 2 )) φ |n|

Tools
Correlation coefficient
• Product moment CC
○ Pearson CC
○
○
For straight line relationship ρ is plus or minus one
If ρ = 0, X and Y are uncorrelated
• Doesn’t imply independence unless X and Y have a bivariate distribution
○ i.e. for n paired observations the sample correlation coefficient is r = S xy
/(S xx
S yy
) 1/2
• S xy
is the sum of the products of deviations of x i
and y i
from their means
• S xx
and S yy
are the sums of squares of deviations from their means
• r =1 implies a direct relationship, r = -1 implies an inverse one
• r = 0 implies virtually no linear relationship, but there may be some other association
- i.e. Points scattered around the circumference of a circle
• Spearman CC
○
2 sets of paired ranks
• Kendall CC
○
2 sets of orderings of the same objects
• Biserial CC
○
y may only take one of 2 variables

Tools (Continued)
F – tests (F distribution)
• Used in analysis of variance to test the Null hypothesis that 2
components estimate the same variance against the alternative
that the numerator estimates a greater variance, the latter
indicating a high “F-value”.
• May also be used to test the acceptability that 2 samples are from
normal distributions with the same variance
• i.e. the ratio of explained variance vs. unexplained variance, OR
between group variability vs. within group variability
Variance ratio
• The ratio of 2 estimates of variance with degrees
of freedom f 1
and f 2
.
• If they both estimate the same variance the ratio will
have a F-distribution with said degrees of freedom.

In case you forgot or missed that day in statistics
F distribution
• The distribution of 2 independent chi-squared variables
Chi-squared distribution
• The sum of squares of n independent standard normal variables (ones that
follow a normal or Gaussian distribution) has a chi-squared (χ 2 )
distribution with n degrees of freedom, a mean n, and a variance 2n.
• Chi-squared test
○ If a value x is expected to occur E i
times for that distribution occurs O i
times.
○ This distribution has n – p degrees of freedom with p being the number of
distribution parameters estimated from the data and used to compute E i
○ When individual distributions are available THEY SHOULD NOT be
grouped simply to allow the test to be applied because the outcome of
the test is NOT independent of the choice of class intervals for grouping.
• i.e. Some grouping may lead to a significant value while others may
not.

F-Tests – on the difficulty of
finding an appropriate time-lag
1) Most models are fundamentally dependant
on unit of time and observational interval.
2) If decision intervals do not coincide with
sampling intervals, temporal aggregation bias
can result.
3) If interval is not fine enough, two correlated
time series may exhibit bi-directional Granger
causality.
4) Continuous time models may provide a better
alternative to discrete time models unless
points in time are specifically chosen for
significance.
5) It is important to try to distinguish between
“true” and “spurious” causality.

is a two variable VAR model broken down into matrix form
Note the matrices A are coefficients estimated by ordinary
least squares
Vector Autoregression –
Captures the evolution and interdependencies between
multiple time series – (Generalizes the Autoregressive
model)
K variables over sample period (t= 1, …, T)
Y t is a k 1 vector
c is a k 1 vector of constants
A i is a k k matrix (for every i=1, …, p)
e t is an error term
For example:

Vector Autoregressions
Macroeconometricians tend to do 4 things
• Macroeconomic forecasts
• Describe and organize data
• Quantify what we do or do not know
• Advise policy
VAR has proved to be powerful and somewhat
reliable for the first 2
Generally there are 3 kinds of VARs
• Reduced form
• Recursive
• Structural

Reduced form VARs
Expresses each variable as a linear function of:
• It’s own past values
• The past values of all other variables considered
• A serially uncorrelated error term
Each equation is estimated by OLS (Ordinary least squares)

Recursive VARs
Each error term is constructed in each regression equation
so as to be uncorrelated with the error in the preceding
equations
• Done by carefully including contemporaneous values as regressors.
Estimation of each equation by OLS results in residuals that
are uncorrelated across equations
• i.e. estimating the reduced form, the computing the Cholesky
factorization of the reduced form VAR covariance matrix
Results depend on the order of the variables
• Changing the order results in changing the VAR equations, coefficients,
and residuals.
• There are n! recursive VARs representing all the orderings

Structural VAR
Uses economic theory for links between variables
Require identifying assumptions that allow
correlations to be interpreted causally.
• Could involve the entire VAR or just a single equation
• The resulting variables allow contemporaneous links to be
estimated using instrumental variables regression
• i.e. – The Taylor rule
○ Considered to be “backward looking” (Federal reserve reacts
to past information to determine federal funds rate)
○ Variables can adjusted by using forecasts from the reduced
form of VAR, resulting in a “forward looking” scenario

Continuous VAR
Can correct for distortional effects of temporal
aggregation without needing to posit a definite
time unit
Provides a basis for importing causality
restrictions to observed discrete data
independently of sampling interval
Obtains efficient estimates of the structural
parameters of the model

Testing
Reduced and recursive forms are used to summarize the comovements
of those 3 series.
Reduced form VAR is used to forecast the variables
• Performance compared to alternate benchmark models
The 2 different structural VARs are used to estimate policy changes
and their repercussions
Data Description
VAR analysis uses reports from Granger-causality tests,
impulse responses, and forecast error variance
decompositions.
Computed automatically by various software
Because the dynamics of VARs are so complicated, these
statistics are more informative than the estimated VAR
regression coefficients, which are usually go unreported.

Performance
VARs capture comovements that cannot be detected in
univariate or bivariate models, because VARs involve current
and lagged values of multiple time series
The standard VAR summary statistics (Granger causality tests,
impulse response functions, etc.) are widely used for showing
these movements.
Limitations
• Standard methods of statistical inference may give misleading
results if some variables are highly persistent.
• Without modification, standard VARs miss nonlinearities, conditional
heteroskedasticity, and drifts or breaks in parameters.
• Small VARs of 2 or 3 variables, are often unstable
• Adding variables increases the number of VAR parameters
○
A 9 variable, 4-lag VAR has 333 unknown coefficients.

Granger Causality and a sampling of
economic processes
Why is capital flowing out of China?
International linkages of Chinese futures
markets
Interaction among China-related stocks:
evidence from a causality test with a new
procedure
Causality and cointegration of stock markets
among the United States, Japan, and the
South China growth circle

Ljungwall, C. Wang, Z. Why is capital flowing out of China? China
Economic Review, 19 (2008) 359-372.

Hua, R. Chen, B. International linkages of Chinese
futures markets. Applied Financial Economics, 17 (2007)
1275-1287.

McCrorie, R. Chambers, M. Granger causality and the sampling of
economic processes. Causality and exogeneity in econometrics,
April 2004.

Huang, B. Yang, C. Causality and cointegration of stock markets among the United
States, Japan, and the south China growth triangle. International review of
financial analysis, 9:3 (2000) 281-297

Tian, G. Wan, G. Interaction among China-related stocks: evidence from a
causality test with a new procedure. Applied Financial Economics, 2004, 14,
67-72

Applications of Granger Causality
What is it used for?
Application 1 – Economics!
Stock prices
Exchange rates
Bonds
Futures
Capital flows
Currency
Interest
Trading volume
Co-integration

Applications of Granger Causality
What is it used for?
Applications 2 – Neuroscience!
Brain mapping
Sensory-neural response
Tracking interdependencies
Researching brain disease Other applications –
human – climate
Political relationships
Sociological interactions
Performance and
revenue in sports
Policy effectiveness

Eview software
Eviews can be useful include: scientific
data analysis and evaluation, financial
analysis macroeconomic forecasting,
simulation, sales forecasting, and cost
analysis.

Microfit software
Microfit has impressive capabilities for data
processing, file management graphical display,
estimation, hypothesis testing and forecasting. It’s
an idean tool for teaching and researching
econometric modelling of time series for
undergraduates and postgraduates.

Jmulti software
Jmulti is meant to be used for empirical data
analysis. When the data are read in, Jmulti
automatically identifies dummy and trend
variables as deterministic

Granger causality is a statistical concept
base on prediction. If lagged values of X
help predict current values of Y in a
forecast formed from lagged values of
both X and Y then X is said to Granger
cause Y.
An application – “Which came first – the
chicken or the egg?”
The chicken
or the egg
Because sometimes Granger Causality
Is more than correlation; It’s causation